For \(d\ge 3\) , a celebrated construction of Burling [1]} shows that the chromatic number of intersection graphs of boxes in \(\mathbb {R}^d\)  cannot be bounded by the clique number alone. The so called Burling graphs provide an infinite family of graphs \(G\)  that are triangle triangle-free, and if \(G\)  has \(n\)  vertices, then \(\chi (G)=\Theta (\log \log n)\) . See [2]} for an alternative construction of a family of \(K_5\) -free intersection graphs of boxes in \(\mathbb {R}^{3}\) , whose chromatic number achieves a similar growths. Let \(G\)  be an intersection graph of \(n\)  boxes in \(\mathbb {R}^d\)  and let \(\omega =\omega (G)\) . Just as before, the best known upper bound on the chromatic number of \(G\)  is supplied by a divide-and-conquer argument (folklore). Therefore, using the result of [3]} as base case, we get \(\chi (G)\le O(\omega )\cdot (\log \omega )(\log n)^{d-2}\) . Here, we provide the first construction which matches this bound up to a factor of \(O((\log w)(\log \log n)^{d-2})\)  for a wide range of values of \(\omega \) .
